Interaction: Fisher’s Optimal Scores Revisited

Abstract

The normal way of fitting a bilinear model to values of a quantitative variable classified by rows and columns is first to fit the main effects of rows and columns and then fit a biadditive model to the residual table. This gives the ordinary least squares optimal fit, usually visualised by biplots and, among other applications areas, is routinely used in genotype $\times$ environment trials. When the dependent variable is categorical, Fisher (1938) proposed scoring the categories by maximising the sum-of-squares of the row and column main effects relative to the residual sum-of-squares; then, the usual biadditive analysis again becomes available, although not used by Fisher. In this paper, it is shown that with categorical variables Fisher could have derived scores by maximising the sum-of-squares of the biadditive part of the model relative to the residual sum-of-squares. With Fisher’s data, this would have three advantages: (i) a better fitting model, (ii) a more parsimonious model and (iii) novel and more powerful visualisation techniques for expressing interactions. Other ways of partitioning the contributions from main effects and biadditivity are also discussed. The results show how small changes in model construction can have profound effects on interpretation. Potential applications extend far beyond the data used by Fisher to whenever the categorical nature of the dependent variable offers the freedom to generate quantifications with generalised interaction parameterisations. Supplementary materials accompanying this paper appear on-line.